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Effect of modeling method on prediction of cortical bone strength adaptation under various loading conditions

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Abstract

The ability to numerically simulate the effects of different loading conditions on the strength adaptation of a bone can be a valuable tool in understanding the relationship between the strength of a bone and its mechanical environment. Because significant strength changes may result from alterations in the profile of the surface of cortical bone, many computational models of bone strength adaptation have been developed to predict load-induced shape modifications. To gain insight into the effects of the modeling methods used for these predictions, the resulting changes to the surface profile of an initially circular cylinder were compared for a number of computational modeling methods. Models based on strain tensor, von Mises stress, and strain energy density were examined under various loading conditions including axial, bending, torsional, and surface forces as well as combinations of these basic loading modes. The differences between the use of a singular reference value and the use of a range of reference values to drive the magnitude of the local shape changes were investigated. Trends in the strain distributions were analyzed. The comparisons performed indicated that, despite the high sensitivity to the values of the model parameters used under the applied loading modes, with the proper selection of these parameters, the diverse methods studied yielded quite similar predictions of the bone’s shape changes, and thus, strength adaptations.

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Abbreviations

\(\vec{\mathbf{F}}_{i}\) :

Dimensionless force vector at node i in “spring-based” smoothing method, Eq. (2a)

\(\vec{\mathbf{x}}_{i}\) :

Location of node i, Eq. (2b)

\(\Delta\vec{\mathbf{x}}_{i}\) :

Displacement at node i in spring smoothing method, Eq. (2a)

\(\Delta\vec{\mathbf{x}}_{j}\) :

Displacement at node j in spring smoothing method, Eq. (2a)

k ij :

Effective spring constant for element edge connecting nodes i and j in spring smoothing method. Dimensions of 1/[L], Eq. (2a)

d ij :

Distance between nodes i and j, Eq. (4)

diff i :

Difference between strain at node i and reference strain for node i, Eq. (3a)

DINFL :

Characteristic distance between nodes over which the effect of the growth at one node on the growth at other nodes is scaled by 1/e, Eq. (4)

G i :

Growth at node i, Eq. (3a)

NW :

Circumferential smoothing weighting factor, Eq. (4)

W axial :

Axial smoothing weighting factor, Eq. (6)

W rate :

Rate factor that scales the growth per load application iteration, Eq. (3a)

i,j,k,n :

indices

ε i (n):

nth strain tensor component for node i. Strain tensor defined as engineering strain transformed into the local cylindrical coordinates for each node i

ε refmax(n):

Max value of reference range for nth strain tensor component

ε refmin(n):

Min value of reference range for nth strain tensor component

\(\sigma_{\mathit{VM}}^{\mathit{iAVG}}\) :

Numerical average of the nodal von Mises stresses of the nodes located on the same surface (inner or outer) as node i that have the same local z-coordinate as node i, Eq. (6)

\(\sigma_{\mathit{VM}}^{\mathit{midAVG}}\) :

Numerical average of the nodal von Mises stresses of nodes located on same surface as node i (inner or outer) and at the midplane (local z-coordinate of this set of nodes = local z max−local z min) of the cylindrical geometry studied, Eq. (6)

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Acknowledgements

This work was partially supported by an Amelia Earhart Fellowship from the Zonta International Foundation.

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  1. Department of Mechanical and Industrial Engineering, New Jersey Institute of Technology, Newark, NJ, 07102, USA

    C. S. Florio & K. A. Narh

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  1. C. S. Florio
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  2. K. A. Narh
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Correspondence to C. S. Florio.

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Florio, C.S., Narh, K.A. Effect of modeling method on prediction of cortical bone strength adaptation under various loading conditions. Meccanica 48, 393–413 (2013). https://doi.org/10.1007/s11012-012-9609-3

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